Simplify and expand the following expression: $ \dfrac{5}{4y - 20}+ \dfrac{1}{5y + 50}+ \dfrac{4}{y^2 + 5y - 50} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $4$ out of denominator in the first term: $ \dfrac{5}{4y - 20} = \dfrac{5}{4(y - 5)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{1}{5y + 50} = \dfrac{1}{5(y + 10)}$ We can factor the quadratic in the third term: $ \dfrac{4}{y^2 + 5y - 50} = \dfrac{4}{(y - 5)(y + 10)}$ Now we have: $ \dfrac{5}{4(y - 5)}+ \dfrac{1}{5(y + 10)}+ \dfrac{4}{(y - 5)(y + 10)} $ The least common multiple of the denominators is: $ 20(y - 5)(y + 10)$ In order to get the first term over $20(y - 5)(y + 10)$ , multiply by $\dfrac{5(y + 10)}{5(y + 10)}$ $ \dfrac{5}{4(y - 5)} \times \dfrac{5(y + 10)}{5(y + 10)} = \dfrac{25(y + 10)}{20(y - 5)(y + 10)} $ In order to get the second term over $20(y - 5)(y + 10)$ , multiply by $\dfrac{4(y - 5)}{4(y - 5)}$ $ \dfrac{1}{5(y + 10)} \times \dfrac{4(y - 5)}{4(y - 5)} = \dfrac{4(y - 5)}{20(y - 5)(y + 10)} $ In order to get the third term over $20(y - 5)(y + 10)$ , multiply by $\dfrac{20}{20}$ $ \dfrac{4}{(y - 5)(y + 10)} \times \dfrac{20}{20} = \dfrac{80}{20(y - 5)(y + 10)} $ Now we have: $ \dfrac{25(y + 10)}{20(y - 5)(y + 10)} + \dfrac{4(y - 5)}{20(y - 5)(y + 10)} + \dfrac{80}{20(y - 5)(y + 10)} $ $ = \dfrac{ 25(y + 10) + 4(y - 5) + 80} {20(y - 5)(y + 10)} $ Expand: $ = \dfrac{25y + 250 + 4y - 20 + 80}{20y^2 + 100y - 1000} $ $ = \dfrac{29y + 310}{20y^2 + 100y - 1000}$